// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H
#define EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H

#include "./Tridiagonalization.h"

namespace Eigen
{

/** \eigenvalues_module \ingroup Eigenvalues_Module
 *
 *
 * \class GeneralizedSelfAdjointEigenSolver
 *
 * \brief Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen3 problem
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the
 * eigendecomposition; this is expected to be an instantiation of the Matrix
 * class template.
 *
 * This class solves the generalized eigenvalue problem
 * \f$ Av = \lambda Bv \f$. In this case, the matrix \f$ A \f$ should be
 * selfadjoint and the matrix \f$ B \f$ should be positive definite.
 *
 * Only the \b lower \b triangular \b part of the input matrix is referenced.
 *
 * Call the function compute() to compute the eigenvalues and eigenvectors of
 * a given matrix. Alternatively, you can use the
 * GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
 * constructor which computes the eigenvalues and eigenvectors at construction time.
 * Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues()
 * and eigenvectors() functions.
 *
 * The documentation for GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
 * contains an example of the typical use of this class.
 *
 * \sa class SelfAdjointEigenSolver, class EigenSolver, class ComplexEigenSolver
 */
template<typename _MatrixType>
class GeneralizedSelfAdjointEigenSolver : public SelfAdjointEigenSolver<_MatrixType>
{
    typedef SelfAdjointEigenSolver<_MatrixType> Base;

public:
    typedef _MatrixType MatrixType;

    /** \brief Default constructor for fixed-size matrices.
     *
     * The default constructor is useful in cases in which the user intends to
     * perform decompositions via compute(). This constructor
     * can only be used if \p _MatrixType is a fixed-size matrix; use
     * GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.
     */
    GeneralizedSelfAdjointEigenSolver()
        : Base()
    {}

    /** \brief Constructor, pre-allocates memory for dynamic-size matrices.
     *
     * \param [in]  size  Positive integer, size of the matrix whose
     * eigenvalues and eigenvectors will be computed.
     *
     * This constructor is useful for dynamic-size matrices, when the user
     * intends to perform decompositions via compute(). The \p size
     * parameter is only used as a hint. It is not an error to give a wrong
     * \p size, but it may impair performance.
     *
     * \sa compute() for an example
     */
    explicit GeneralizedSelfAdjointEigenSolver(Index size)
        : Base(size)
    {}

    /** \brief Constructor; computes generalized eigendecomposition of given matrix pencil.
     *
     * \param[in]  matA  Selfadjoint matrix in matrix pencil.
     *                   Only the lower triangular part of the matrix is referenced.
     * \param[in]  matB  Positive-definite matrix in matrix pencil.
     *                   Only the lower triangular part of the matrix is referenced.
     * \param[in]  options A or-ed set of flags {#ComputeEigenvectors,#EigenvaluesOnly} | {#Ax_lBx,#ABx_lx,#BAx_lx}.
     *                     Default is #ComputeEigenvectors|#Ax_lBx.
     *
     * This constructor calls compute(const MatrixType&, const MatrixType&, int)
     * to compute the eigenvalues and (if requested) the eigenvectors of the
     * generalized eigenproblem \f$ Ax = \lambda B x \f$ with \a matA the
     * selfadjoint matrix \f$ A \f$ and \a matB the positive definite matrix
     * \f$ B \f$. Each eigenvector \f$ x \f$ satisfies the property
     * \f$ x^* B x = 1 \f$. The eigenvectors are computed if
     * \a options contains ComputeEigenvectors.
     *
     * In addition, the two following variants can be solved via \p options:
     * - \c ABx_lx: \f$ ABx = \lambda x \f$
     * - \c BAx_lx: \f$ BAx = \lambda x \f$
     *
     * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.cpp
     * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.out
     *
     * \sa compute(const MatrixType&, const MatrixType&, int)
     */
    GeneralizedSelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB, int options = ComputeEigenvectors | Ax_lBx)
        : Base(matA.cols())
    {
        compute(matA, matB, options);
    }

    /** \brief Computes generalized eigendecomposition of given matrix pencil.
     *
     * \param[in]  matA  Selfadjoint matrix in matrix pencil.
     *                   Only the lower triangular part of the matrix is referenced.
     * \param[in]  matB  Positive-definite matrix in matrix pencil.
     *                   Only the lower triangular part of the matrix is referenced.
     * \param[in]  options A or-ed set of flags {#ComputeEigenvectors,#EigenvaluesOnly} | {#Ax_lBx,#ABx_lx,#BAx_lx}.
     *                     Default is #ComputeEigenvectors|#Ax_lBx.
     *
     * \returns    Reference to \c *this
     *
     * Accoring to \p options, this function computes eigenvalues and (if requested)
     * the eigenvectors of one of the following three generalized eigenproblems:
     * - \c Ax_lBx: \f$ Ax = \lambda B x \f$
     * - \c ABx_lx: \f$ ABx = \lambda x \f$
     * - \c BAx_lx: \f$ BAx = \lambda x \f$
     * with \a matA the selfadjoint matrix \f$ A \f$ and \a matB the positive definite
     * matrix \f$ B \f$.
     * In addition, each eigenvector \f$ x \f$ satisfies the property \f$ x^* B x = 1 \f$.
     *
     * The eigenvalues() function can be used to retrieve
     * the eigenvalues. If \p options contains ComputeEigenvectors, then the
     * eigenvectors are also computed and can be retrieved by calling
     * eigenvectors().
     *
     * The implementation uses LLT to compute the Cholesky decomposition
     * \f$ B = LL^* \f$ and computes the classical eigendecomposition
     * of the selfadjoint matrix \f$ L^{-1} A (L^*)^{-1} \f$ if \p options contains Ax_lBx
     * and of \f$ L^{*} A L \f$ otherwise. This solves the
     * generalized eigenproblem, because any solution of the generalized
     * eigenproblem \f$ Ax = \lambda B x \f$ corresponds to a solution
     * \f$ L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) \f$ of the
     * eigenproblem for \f$ L^{-1} A (L^*)^{-1} \f$. Similar statements
     * can be made for the two other variants.
     *
     * Example: \include SelfAdjointEigenSolver_compute_MatrixType2.cpp
     * Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType2.out
     *
     * \sa GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
     */
    GeneralizedSelfAdjointEigenSolver& compute(const MatrixType& matA, const MatrixType& matB, int options = ComputeEigenvectors | Ax_lBx);

protected:
};


template<typename MatrixType>
GeneralizedSelfAdjointEigenSolver<MatrixType>& GeneralizedSelfAdjointEigenSolver<MatrixType>::
    compute(const MatrixType& matA, const MatrixType& matB, int options)
{
    eigen_assert(matA.cols() == matA.rows() && matB.rows() == matA.rows() && matB.cols() == matB.rows());
    eigen_assert((options & ~(EigVecMask | GenEigMask)) == 0 && (options & EigVecMask) != EigVecMask && ((options & GenEigMask) == 0 || (options & GenEigMask) == Ax_lBx || (options & GenEigMask) == ABx_lx || (options & GenEigMask) == BAx_lx) && "invalid option parameter");

    bool computeEigVecs = ((options & EigVecMask) == 0) || ((options & EigVecMask) == ComputeEigenvectors);

    // Compute the cholesky decomposition of matB = L L' = U'U
    LLT<MatrixType> cholB(matB);

    int type = (options & GenEigMask);
    if ( type == 0 )
        type = Ax_lBx;

    if ( type == Ax_lBx ) {
        // compute C = inv(L) A inv(L')
        MatrixType matC = matA.template selfadjointView<Lower>();
        cholB.matrixL().template solveInPlace<OnTheLeft>(matC);
        cholB.matrixU().template solveInPlace<OnTheRight>(matC);

        Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);

        // transform back the eigen3 vectors: evecs = inv(U) * evecs
        if ( computeEigVecs )
            cholB.matrixU().solveInPlace(Base::m_eivec);
    }
    else if ( type == ABx_lx ) {
        // compute C = L' A L
        MatrixType matC = matA.template selfadjointView<Lower>();
        matC            = matC * cholB.matrixL();
        matC            = cholB.matrixU() * matC;

        Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);

        // transform back the eigen3 vectors: evecs = inv(U) * evecs
        if ( computeEigVecs )
            cholB.matrixU().solveInPlace(Base::m_eivec);
    }
    else if ( type == BAx_lx ) {
        // compute C = L' A L
        MatrixType matC = matA.template selfadjointView<Lower>();
        matC            = matC * cholB.matrixL();
        matC            = cholB.matrixU() * matC;

        Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);

        // transform back the eigen3 vectors: evecs = L * evecs
        if ( computeEigVecs )
            Base::m_eivec = cholB.matrixL() * Base::m_eivec;
    }

    return *this;
}

}   // end namespace Eigen

#endif   // EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H
